Malcev, protomodular, homological and semi-abelian categories

This entry is about the book:

on nonabelian homological algebra, such as its diagram chasing lemmas, in Mal'cev categories, protomodular categories, semi-abelian categories and homological categories.

From the introduction:

The most striking successes of category theory, as far as clarification of mathematical situations is concerned, are probably the theory of abelian categories and the theory of toposes. This is not too amazing since both theories are closely related to the development of sheaf theory, a context in which it is desirable to get rid of the usual notion of element.

But up to recently, category theory did not provide any comparable insight in General Algebra, a domain in which element-based mathematics remains the slogan. In particular, category theory could not provide a structural tool able to grasp, even in the most representative category of classical algebra - namely, the category Gp \mathrm{Gp} of groups - the deep essence of the notion of normal subobject: namely, an equivalence class for a congruence and not just the kernel of a morphism.

And category theory could not grasp either the conceptual foundations of the homological lemmas: the Nine Lemma, the Snake Lemma, which remain valid and strongly meaningful in the category Gp\mathrm{Gp} of groups, even if this category does not belong to the abelian setting in which these lemmas are generally proved in a significant categorical way.

Of course, there have been since a long time attempts to provide an axiomatic context in which to get the isomorphism theorems, the decomposition theorems or the previous homological lemmas for the varieties of Universal Algebra: Baer (1947, [6]), Goldie (1952, [48]), Atiyah (1956, [5]), Higgins (1956, [54]), Kurosh (1959, [73]), Hilton-Ledermann (1960, [55]), Eckmann-Hilton (1962, [40]), Tsalenko (1967, [92]), but also Hofmannn (1960, [56]), Fröhlich (1961, [46]), Huq (1968, [57]), Gerstenhaber (1970, [47]), Burgin (1970, [34]), Orzech (1972, [83]).

These first attempts, despite their interest, consist generally in a long list of axioms whose independence is certainly not clear. But more importantly, these axioms look desperately heavy and complicated in comparison with the elegance of the characterization of abelian models. We refer the reader to the introduction of the paper by Janelidze-Márki-Tholen (2002, [60]) for a reliable historical approach to this topic. … Establishing an organic and synthetic connection between all these attempts the ambition of this book. To achieve this, an additional ingredient was necessary, of purely categorical nature: the fibration of points. This fibration allows representing every category as a fibration whose fibres are pointed categories, i.e. categories with a zero object (see Bourn, 1996, [17]). This book will give evidence that the fibration of points emphasizes the importance of split epimorphisms in the context of algebraic theories, but also that this fibration of points has a very strong classification power: see on page 466 the table summarizing these classification properties. …


(with links to the related nnlab entries)

  • Preface
  1. Metatheorems

  2. Mal'cev categories

  3. Protomodular categories

  4. Homological categories

  5. Semi-abelian categories

  6. Strongly protomodular categories

  7. Essentially affine categories


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